One simple way to explain the stimulated emission is to say that the photon present in the initial state does nothing and it is the indistinguishability of the two photons, in the final state (note that one  was incident and one more has been added by emission process) which is responsible for stimulated emission. Since we do not know which one of the final two photons is the new photon, any one can be the new photon and the emission probability is thus double of that for spontaneous emission and it happens due to presence of one photon initially.

Continuing the argument, we can say that if n identical photon were present in any mode, the probability of emission of a photon into this mode shall be n+1 times that of spontaneous emission. The addition probability which is n times that for spontaneous emission is the stimulated emission.

The same conclusion is obtained by quantum mechanics, as the matrix element for the emission process involves creation operator between bra for n+1 photons and ket for n photons, which is equal to (n+1)1/2. Thetransition probability which is proportional to square of the matrix element is obviously proportional to n+1

Source: http://physics.stackexchange.com/questions/65711/why-is-the-photon-emitted-in-the-same-direction-as-incoming-radiation-in-laser

The word "stimulated" means that the emission of the photon is "encouraged" by the existence of photons in the same state as the state where the new photon may be added. The "same state" is one that has the same frequency, the same polarization, and the same direction of motion. Such a one-photon state may be described by the wave vector and the polarization vector, e.g. |k⃗ ,λ⟩.

The physical reason why photons like to be emitted in the very same state as other photons is that they are bosons obeying the Bose-Einstein statistics. The probability amplitude for a new, (N−1)-st photon to be added into a one-photon state which already has N photons in it is proportional to the matrix element of the raising operator

⟨N+1|a†|N⟩=N+1−−−−−√

of the harmonic oscillator between the N times and (N+1) times excited levels. Because the probability amplitude scales like N+1−−−−−√, the probability for the photon to be emitted into the state goes like the squared amplitude i.e. as N+1. Recall that N is the number of photons that were already in that state.

This coefficient N+1 may be divided to 1 plus N. The term 1 describes the probability of a spontaneous emission – that occurs even if no other photons were present in the state to start with – while the term N is the stimulated emission whose probabilities scales with the number of photons that are already present.

But in all cases, we must talk about "exactly the same one-photon state" which also means that the direction of the motion is the same. It's because quantum field theory associates one quantum harmonic oscillator with each state i.e. with each information k⃗ about the direction of motion and wavelength; combined with a binary information about λ, the polarization (e.g.left-handed vs right-handed).

Dear Prakash, Dear Charles and Dear Poojary;

Thank you for your kindly informations and deeply helps. I am clarified right now. That was an information that I could not find on the internet for a long time.

Best Regards.

I shall try to explain it with simple analogy.

Every energy state is associated with certain frequency (wave equation) .

Stimulation is a quantum phenomenon (statistical)

but can be explained with wave theory.

For interaction the resonance of two energies is essential.

After the inversion many electrons assume a quasi stable state (still oscillating with corresponding frequency). when incoming light wave encounters quasi stable electron,

Lossless interaction between photon and electron would take place only when both frequency and phase of each are same means coherent. 

Hello Cagil. These links might help you in getting more insight

https://spie.org/publications/fg12_p02_spontaneous_and_stimulated_emission?SSO=1

https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Book%3A_Direct_Energy_(Mitofsky)/07%3A_Lamps%2C_LEDs%2C_and_Lasers/7.01%3A_Absorption%2C_Spontaneous_Emission%2C_Stimulated_Emission

https://physics.stackexchange.com/questions/65711/why-is-the-photon-emitted-in-the-same-direction-as-incoming-radiation-in-laser